ON THE MOTION OF PENDULUMS. 73 time. Suppose first that X is two inches, and t ten seconds. Then lOwy^"2 = 1/256, and c : C0 :: 1 : 0*2848, so that the height of the waves, which varies as c, is only about a quarter of what it was. Accordingly, the ripples excited on a small pool by a puff of wind rapidly subside when the exciting cause ceases to act. Now suppose that \ is 40 fathoms or 2880 inches, and that t is 86400 seconds or a whole day. In this case 167r2//£X~2 is equal to only 0*005232, so that by the end of an entire day, in which time waves of this length would travel 574 English miles, the height would be diminished by little more than the one two-hundredth part in consequence of friction. Accordingly, the long swells of the ocean are but little allayed by friction, and at last break on some shore situated at the distance of perhaps hundreds of miles from the region where they were first excited. 52. It is worthy of remark, that in the case of a homogeneous incompressible fluid, whenever udx + vdy + wdz is an exact differential, not only are the ordinary equations of fluid motion satisfied*, but the equations obtained when friction is taken into account are satisfied likewise. It is only the equations of condition which belong to the boundaries of the fluid that are violated. Hence any kind of motion which is possible according to the ordinary equations, and which is such that udx + vdy + wdz is an exact differential, is possible likewise when friction is taken into account, provided we suppose a certain system of normal and tangential pressures to act at the boundaries of the fluid, so as to satisfy the equations of condition. The requisite system of pressures is given by the system of equations (133). Since p, disappears from the general equations (1), it follows thatjp is the same function as before. But in the first case the system of pressures at the surface was P1~P2 = P3=p) T1 = T2=2\ = 0. Hence if APX &c. be the additional pressures arising from friction, we get from (133), observing that 8=0, and that udx 4- vdy -f wdz is an exact differential d<f>, (112), * It is here supposed that the forces X, Y, Z are such that Xdx + Ydy + Zdz is an exact differential.